When Sitecore indicates that publishing has finished (via normal publish dialog) does this mean that subsequent indexing will also have finished, or does the dialog only report on publish status?
By default web indexes uses
It's executed asynchronously and depending on your application settings, can be distributed to all CD servers (e.g. with Lucene).
That's why the dialog only reports the publishing itself.
If you want to see if the reindexing is completed, you may want to try
/sitecore/admin/jobs.aspx page. But those are only jobs executed on that particular server you're browsing, so it will not tell you if other servers still process their indexes.
Sitecore calls by default web indexes uses onPublishEndAsync strategy. But there are different strategy which you need to check in your case which you can find in Sitecore.ContentSearch.Lucene.DefaultIndexConfiguration.config or Sitecore.ContentSearch.Solr.DefaultIndexConfiguration.config
Then you will get the whole idea if it got published or not.
you can also monitor the jobs which are running in Sitecore as mentioned by Matthew Dresser in above answer.